Let
be a lattice (or a bounded lattice or a complemented lattice, etc.), and let
be the covering relation of :

Then
is locally realized provided that for every finite subset of , there is a finitely generated sublattice of , that contains , for which . It can be shown that is locally realized if and only if there is a hyperfinitely
generated sublattice of such that . Using this characterization
of locally realized covering relations, the following standard result can be proved
using nonstandard methods:

Let
be a locally finite lattice in which the covering relation is locally realized, and
let
be the sublattice of which is generated by . Then is a connected tolerance of , and it is in fact the smallest locally subconnected (and
locally connected) tolerance of .